Number of palindromic (in base 2) quadruples that sum to n.

1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 6, 4, 7, 4, 7, 6, 9, 6, 11, 6, 12, 7, 13, 7, 16, 7, 15, 9, 17, 7, 19, 9, 20, 10, 22, 10, 26, 11, 25, 12, 27, 11, 30, 11, 29, 12, 30, 10, 35, 11, 32, 14, 36, 11, 39, 13, 36, 13, 38, 11, 42, 12, 39, 15, 42, 13, 48, 14, 47, 17, 47, 13

0

Note the the terms for 2*n are about the square of terms for 2*n+1. See S000707.

T. D. Noe, Plot of 10000 terms

T. D. Noe, Table of 10000 terms

Yu Gao, Represent a natural number as the sum of palindromes in various bases, arXiv 1508.06185 (Aug 24 2015)

(Mma) base = 2; mx = 100; pal = Select[Range[0, mx], (d = IntegerDigits[#, base]; d == Reverse[d]) &]; Table[Length[IntegerPartitions[n, {4}, pal]], {n, 0, mx}]

nonn,base

T. D. Noe, Sep 23 2015